CONTENTS

Preface to the Enlarged and Corrected Printing............... V

Preface. ,..,.***..-.....................ix

Notations..............................XV

Chapter I

ELEMENTS OF THE THEORY OF SETS.............. 1

1. Elements and sets. 2. Boolean algebra. 3. Product of two sets. 4. Mappings.
5. Direct and inverse images. 6. Surjective, injective, and bijective mappings.
7. Composition of mappings. 8. Families of elements. Union, intersection, and
products of families of sets. Equivalence relations. 9. Denumerable sets.

Chapter 11

REAL NUMBERS........................ 16

1. Axioms of the real numbers. 2. Order properties of the real numbers. 3. Least
upper bound and greatest lower bound.

Chapter 111

METRIC SPACES........................ 27

1. Distances and metric spaces. 2. Examples of distances. 3. Isornetries.
4. Balls, spheres, diameter. 5. Open sets. 6. Neighborhoods. 7. Interior of
a set. 8. Closed sets, cluster points, closure of a set. 9. Dense subsets; separable
spaces. 10. Subspaces of a metric space. 11. Continuous mappings. 12. Homeo-
morphisms. Equivalent distances. 13. Limits. 14. Cauchy sequences, complete
spaces. 15. Elementary extension theorems. 16. Compact spaces. 17. Compact
sets. 18. Locally compact spaces. 19. Connected spaces and connected sets.
20. Product of two metric spaces.

Chapter IV

ADDITIONAL PROPERTIES OF THE REAL LINE.......... 79

1. Continuity of algebraic operations. 2. Monotone functions. 3. Logarithms
and exponentials. 4. Complex numbers. 5. The Tietze-Urysohn extension
theorem.rts of